I am trying to find the limiting matrix for the indicated standard form.
$$ \begin{matrix} 1 & 0 & 0 \\ .1 & .6 & .3 \\ .2 & .2 & .6 \\ \end{matrix} $$
I am confused because all of my examples in the text book have the top left section of the matrix as
$$ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \\ \end{matrix} $$ Can anyone elaborate on this for me? Can I just carry on as normal with the matrix the way it is or am I totally misunderstanding how these things work?
Your Markov chain seems to be absorbing: The probability to leave state 1 is zero, hence once you enter state one, you stay there forever. States 2 and 3 can access state 1, so after some sufficiently long time the Markov chain will end up in state 1 (and stay there forever). I would thus guess that the limiting matrix satisfies
$$ \lim_{n\to\infty} P^n = \begin{bmatrix} 1 & 0 & 0 \cr 1 & 0 & 0 \cr 1 & 0 & 0\end{bmatrix} $$